Over-Parameterized Variational Optical Flow Method

ABSTRACT

An optical flow estimation process based on a spatio-temporal model with varying coefficients multiplying a set of basis functions at each pixel. The benefit of over-parameterization becomes evident in the smoothness term, which instead of directly penalizing for changes in the optic flow, accumulates a cost of deviating from the assumed optic flow model. The optical flow field is represented by a general space-time model comprising a selected set of basis functions. The optical flow parameters are computed at each pixel in terms of coefficients of the basis functions. The model is thus highly over-parameterized, and regularization is applied at the level of the coefficients, rather than the model itself. As a result, the actual optical flow in the group of images is represented more accurately than in methods that are known in the art.

FIELD OF THE INVENTION

The present invention relates in general to image processing and inparticular to a method and system for determining optical flow.

BACKGROUND OF THE INVENTION

Optical flow is a concept which approximates the motion of objectswithin a visual representation. Optical flow is the velocity field whichwarps one image into another (usually very similar) image. Optical flowtechniques are based on the idea that the same physical point on anobject in the scene is captured by the camera in corresponding points inthe two images preserving certain image properties such as brightness,the gradient vector, etc.

Optical flow computations are central to many image processingapplications that deal with groups of similar images. For example, imagesequence compression algorithms commonly use optical flow parameters torepresent images compactly in terms of changes relative to preceding orsucceeding images in the sequence. Optical flow parameters are also usedin three-dimensional reconstruction by stereo matching of pixels in agroup of images taken of an object from different angles, or by trackingthe motion of rigid objects in a scene, as well as in image resolutionenhancement. In addition, variations in optical flow over the area of animage may be used in image segmentation and in tracking the motion of anobject across a sequence of images.

Despite much research effort invested in addressing optical flowcomputation it remains a challenging task in the field of computervision. It is a necessary step in various applications like stereomatching, video compression, object tracking, depth reconstruction andmotion based segmentation. Hence, many approaches have been proposed foroptical flow computation. Most methods assume brightness constancy andintroduce additional assumptions on the optical flow in order to dealwith the inherent aperture problem. Lucas and Kanade (1981) tackled theaperture problem by solving for the parameters of a constant motionmodel over image patches. Subsequently, Irani et al. (1993, 1997) usedmotion models in a region in conjunction with Lucas-Kanade in order torecover the camera ego-motion. Spline based motion models were suggestedby Szeliski and Coughlan (1997).

Horn and Schunck (1981) sought to recover smooth flow fields and werethe first to use functional minimization for solving optical flowproblems employing mathematical tools from calculus of variations. Theirpioneering work put forth the basic idea for solving dense optical flowfields over the whole image by introducing a quality functional with twoterms: a data term penalizing for deviations from the brightnessconstancy equation, and a smoothness term penalizing for variations inthe flow field. Several important improvements have been proposedfollowing their work. Nagel (1990, 1986) proposed an oriented smoothnessterm that penalizes anisotropically for variations in the flow fieldaccording to the direction of the intensity gradients. Ari and Sochen(2006) recently used a functional with two alignment terms composed ofthe flow and image gradients. Replacing quadratic penalty terms byrobust statistics integral measures was proposed in (Black and Anandan1996; Deriche et al. 1995) in order to allow sharp discontinuities inthe optical flow solution along motion boundaries. Extensions tomulti-frame formulations of the initial two-frames formulation allowedthe consideration of spatiotemporal smoothness to replace the originalspatial smoothness term (Black and Anandan 1991; Farneback 2001; Nagel1990; Weickert and Schnörr 2001). Brox et al. (2004, 2006) demonstratedthe importance of using the exact brightness constancy equation insteadof its linearized version and added a gradient constancy to the dataterm which may be important if the scene illumination changes in time.Cremers and Soatto (2005) proposed a motion competition algorithm forvariational motion segmentation and parametric motion estimation. Amiazand Kiryati (2005) followed by Brox et al. (2006) introduced avariational approach for joint optical flow computation and motionsegmentation. In Farneback (2000, 2001), a constant and affine motionmodel is employed. The motion model is assumed to act on a region, andoptic flow based segmentation is performed by a region growingalgorithm. In a classical contribution to structure from motion Adiv(1985) used optical flow in order to determine motion and structure ofseveral rigid objects moving in the scene. Sekkati and Mitiche (2006)used joint segmentation and optical flow estimation in conjunction witha single rigid motion in each segmented region. Vázquez et al. (2006)used joint multi-region segmentation with high order DCT basis functionsrepresenting the optical flow in each segmented region.

1. SUMMARY OF THE INVENTION

It is an object of the present invention to represent the optical flowvector at each pixel by different coefficients of the same motion modelin a variational framework. Such a grossly over-parameterizedrepresentation has the advantage that the smoothness term may nowpenalize deviations from the motion model instead of directly penalizingthe change of the flow. For example, in an affine motion model, if theflow in a region can be accurately represented by an affine model, thenin this region there will be no flow regularization penalty, while inthe usual setting there is a cost resulting from the changes in the flowinduced by the affine model. This over-parameterized model therebyoffers a richer means for optical flow representation. For segmentationpurposes, the over-parameterization has the benefit of makingsegmentation decisions in a more appropriate space (e.g. the parametersof the affine flow) rather than in a simple constant motion model space.The work of Ju et al. (1996) is related to our methodology, they usedlocal affine models to describe the motion in image regions imposingspatial smoothness on the affine parameters between neighboring patches.The key and conceptually very important difference is that, in ourapproach, the model is represented at the pixel level which makes theproblem over-parameterized while the patch size chosen in (Ju et al.1996) makes it under-parameterized and requires the choice of aneighborhood size.

In one aspect, the present invention relates to a method of determiningthe optical flow vector of a plurality of image pixels between twoconsecutive images in a group of images, the method comprising the stepsof:

(i) creating multiple sets of the group of images wherein each setcontains a decreasing number of pixels per image;

(ii) defining an optical flow motion model applicable to the group ofimages in terms of a set of basis functions;

(iii) representing the optical flow vector of each pixel image bydifferent coefficients of the same motion model;

(iv) starting with the set of group of images with the lowest number ofpixels per image, determining respective over-parameterized optical flowvectors at the pixels responsively to motion between the images in thegroup by computing respective regularized coefficients of the basisfunctions at each of the pixels;

(v) interpolating the results to the next higher resolution set of groupof images and refining the estimation using the over-parameterizedoptical flow representation; and

(vi) repeating step (v) until the highest resolution set of group ofimages is reached.

The present invention introduces a novel over-parameterized variationalframework for accurately solving the optical flow problem. The flowfield is represented by a general space-time model. The proposedapproach is useful and highly flexible in the fact that each pixel hasthe freedom to choose its own set of model parameters. Subsequently, thedecision on the discontinuity locations of the model parameters isresolved within the variational framework for each sequence. In mostscenarios, the optical flow would be better represented by a piecewiseconstant affine model or a rigid motion model rather than a piecewiseconstant flow. Therefore, compared to existing variational techniques,the smoothness penalty term modeled by the proposed overparameterizationmodels yields better optic flow recovery performance as demonstrated byour experiments. Our experiments focused on spatial basis functions,though the same principles apply for spatio-temporal basis functions.Incorporating learning of the basis functions (dictionaries) forspecific scenes is of great interest and useful for video compression.According to the invention, motion segmentation based on optical flowshould generally be replaced by segmentation in the higher dimensionalparameter space as suggested by our initial results presented herein forthe synthetic sequence. Although the models suggested were allover-parameterized, an under-parameterized model might also be used inthis framework, for example, in case one has prior knowledge regardingconstraints between the u and v components (as in stereo matching orwhen we know the optic flow to be radial).

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A, 1B show the optic flow of the Yosemite sequence, FIG. 1Ashowing the ground truth and FIG. 1B according to one embodiment of theinvention with pure translation model.

FIGS. 2A, 2B, 2C show Images of the angular error, white indicates zeroerror and black indicates an error of 3 degrees or above. FIG. 2A isaccording to one embodiment of the invention method with the affinemodel. FIG. 2B is according to one embodiment of the invention methodwith the pure translation model. FIG. 2C uses the method of (Brox et al.2004).

FIGS. 3A, 3B show a histogram (FIG. 3A) and Cumulative probability (FIG.3B) of the angular errors. Solid lines denote pure translation model;dash dot lines denotes the affine model; and dotted lines denote opticflow obtained in (Brox et al. 2004).

FIGS. 4A, 4B, 4C show the solution of the affine parameters for theYosemite sequence for: A₁ (FIG. 4A), A₂ (FIG. 4B) and A₃. (FIG. 4A).

FIGS. 5A, 5B, 5C show the solution of the affine parameters for theYosemite sequence for: A₄ (FIG. 5A), A₅ (FIG. 5B) and A₆. (FIG. 5A).

FIG. 6A, 6B, 6C show Solution of the pure translation parameters for theYosemite sequence for: A₁ (FIG. 6A), A₂ (FIG. 6B) and A₃. (FIG. 6A).

FIGS. 7A, 7B show two images obtained in a synthetic piecewise constantaffine flow example based on FIGS. 8A, 8B.

FIGS. 8A, 8B show a synthetic piecewise affine flow—ground truth: FIG.8A showing the u component; and FIG. 8B showing the v component.

FIGS. 9A, 9B, 9C show the solution of the affine parameters based onFIGS. 8A, 8B. FIG. 9A showing A₁; FIG. 9B showing A₂; and FIG. 9Cshowing A₃.

FIGS. 10A, 10B, 10C show the solution of the affine parameters based onFIGS. 8A, 8B. FIG. 10A showing A₄; FIG. 10B showing A₅; and FIG. 10Cshowing A₆.

FIGS. 11A, 11B show the ground truth for the synthetic test (FIG. 11A)and computed flow by the affine over-parameterization model (FIG. 11B).

FIGS. 12A, 12B, 12C show an example of a flower garden sequence FIG.12A; and optical flow computed by the 2D affine over-parameterizationmodel, u (FIG. 12B) and v (FIG. 12C).

FIGS. 13A, 13B the optical flow computed by the 2D pure translationover-parameterization model based on the flower garden of FIG. 12A: u(FIG. 13A) and v (FIG. 13B).

FIGS. 14A, 14B, 14C show the flower garden sequence based on FIG. 12A:the solution of the pure translation parameter. FIG. 14A showing A₁;FIG. 14B showing A₂; and FIG. 14C showing A₃.

FIGS. 15A, 15B show two frames of a road sequence.

FIGS. 16A, 16B show the optical flow of the road sequence. u is shown onFIGS. 16A and v is shown on FIG. 16B.

FIGS. 17A, 17B show the difference of the frames. FIG. 17A is thedifference of the original frames (PSNR=23.73 dB). FIG. 17B shows motioncompensated difference by the estimated optical flow (PSNR=38.17 dB).Both images are scaled by the same linear transformation from errors togray levels.

DETAILED DESCRIPTION OF THE INVENTION

In the following detailed description of various embodiments, referenceis made to the accompanying drawings that form a part thereof, and inwhich are shown by way of illustration specific embodiments in which theinvention may be practiced. It is understood that other embodiments maybe utilized and structural changes may be made without departing fromthe scope of the present invention.

2. OVER-PARAMETRIZATION MODEL

We propose to represent the optical flow (u(x, y, t), v(x, y, t)) by thegeneral over-parameterized space-time model

$\begin{matrix}{{{u( {x,y,t} )} = {\sum\limits_{i = 1}^{n}\; {{A_{i}( {x,y,t} )}{\varphi_{i}( {x,y,t} )}}}},} & (1) \\{{{v( {x,y,t} )} = {\sum\limits_{i = 1}^{n}\; {{A_{i}( {x,y,t} )}{\eta_{i}( {x,y,t} )}}}},} & \;\end{matrix}$

where, Φ_(i)(x, y, t) and η_(i)(x, y, t), i=1, . . . , n are n basisfunctions of the flow model, while the ^(A)i are space and time varyingcoefficients of the model. This is an obviously heavilyover-parameterized model since for more than two basis functions, thereare typically many ways to express the same flow at any specificlocation. This redundancy however will be adequately resolved by aregularization assumption applied to the coefficients of the model. Thecoefficients and basis functions may be general functions of space-time,however, they play different roles in the functional minimizationprocess: The basis functions are fixed and selected a priori. Thecoefficients are the unknown functions we solve for in the optical flowestimation process. In our model, appropriate basis functions are suchthat the true flow could be described by approximately piecewiseconstant coefficients, so that most of the local spatio-temporal changesof the flow are induced by changes in the basis functions and not byvariations of the coefficients. This way, regularization applied to thecoefficients (as will be described later on) becomes meaningful sincemajor parts in the optic flow variations can be described withoutchanges of the coefficients. For example, rigid body motion has aspecific optical flow structure which can explain the flow using onlysix parameters at locations with approximately constant depth. Let usstart from conventional optical flow functionals that include a dataterm E_(D) (u, v), that measures the deviation from the brightnessconstancy assumption, and a regularization (or smoothness) term E_(S)(u, v) that quantifies the smoothness of the flow field. The solution isthe minimizer of the sum of the data and smoothness terms

E(u,v)=E _(D)(u,v)+αE _(S)(u,v).  (2)

The main difference between the diverse variational methods is in thechoice of data and smoothness terms, and in the numerical methods usedfor solving for the minimizing flow field (u(x, y, t), v(x, y, t)). Forthe data term we shall use the functional)

E _(D)(u,v)=∫ψ((I(x+w)−I(x))²)dx,  (3)

where, x=(x, y, t)^(T) and w=(u, v, 1)^(T). This is the integral measureused for example in (Brox et al. 2004) (omitting the gradient constancyterm). The function ψ(s²)=√{square root over (s²+ε²)} that is by nowwidely used, induces an approximate L₁ metric of the data term for asmall s. The smoothness term used in (Brox et al. 2004) is given by

E _(S)(u,v)=∫ψ(∥{tilde over (∇)}u∥ ² +∥{tilde over (∇)}v∥ ²)dx  (4)

Where {tilde over (∇)}f≡(f_(x),f_(y),ω_(t)f_(t))^(T) denotes theweighted spatiotemporal gradient. ω_(t) indicates the weight of thetemporal axis relative to the spatial axes in the context of thesmoothness term (ω_(t)=1 is used in (Brox et al. 2004)). Inserting theover-parameterized model into the data term, we have

$\begin{matrix}{{E_{D}( A_{i} )} = {\int{{\Psi ( ( {{I\begin{pmatrix}{{x + {\sum\limits_{i = 1}^{n}\; {A_{i}\varphi_{i}}}},} \\{{y + {\sum\limits_{i = 1}^{n}\; {A_{i}\eta_{i}}}},{t + 1}}\end{pmatrix}} - {I( {x,y,t} )}} )^{2} )}{{x}.}}}} & (5)\end{matrix}$

Our proposed smoothness term replaces (4) with a penalty forspatio-temporal changes in the coefficient functions,

$\begin{matrix}{{E_{S}( A_{i} )} = {\int{{\Psi( {\sum\limits_{i = 1}^{n}{{\overset{\sim}{\nabla}A_{i}}}^{2}} )}{{x}.}}}} & (6)\end{matrix}$

Notice that in (6), constant parameters of the model can describechanges of the flow field according to the chosen model as described in(1) (e.g. Euclidean, affine, etc.) without smoothness penalty, whereasin (4), any change in the flow field is penalized by the smoothnessterm. For the sake of simplicity of the resulting Euler-Lagrangeequations, we have omitted writing explicit relative weights to thedifferent coefficients in the smoothness term, the weighting isalternatively achieved by scaling the basis functions by appropriatefactors as will be shown in the description of the motion models.Scaling a basis function by a small factor would mean that in order toachieve the same overall influence on the optical flow, thecorresponding coefficient would have to make larger changes(proportional to the inverse of the factor). These larger changes wouldbe suppressed by the regularization term. On the other hand, scaling abasis function by a large factor would scale down the changes requiredfrom the corresponding coefficient in order to achieve the same overallchange and therefore would result in less regularization for thisspecific term.

2.1 Euler-Lagrange Equations

For an over-parameterization model with n coefficients, there are nEuler-Lagrange equations, one for each coefficient. The Euler-Lagrangeequation for A_(q) (q=1, . . . , n) is given by

$\begin{matrix}{{{{\Psi^{\prime}( I_{z}^{2} )}{I_{z}( {{I_{x}^{+}\varphi_{q}} + {I_{y}^{+}\eta_{q}}} )}} - {\alpha \cdot {{div}( {{\Psi^{\prime}( {\sum\limits_{i = 1}^{n}{{\overset{\sim}{\nabla}A_{i}}}^{2}} )}{\hat{\nabla}A_{q}}} )}}} = 0.} & (7)\end{matrix}$

where {circumflex over (∇)}f≡(f_(x),f_(y),ω_(t) ²f_(t))^(T), and

$\begin{matrix}{{I_{x}^{+}:={I_{x}( {{x + {\sum\limits_{i = 1}^{n}\; {A_{i}\varphi_{i}}}},{y + {\sum\limits_{i = 1}^{n}\; {A_{i}\eta_{i}}}},{t + 1}} )}},} & (8) \\{{I_{y}^{+}:={I_{y}( {{x + {\sum\limits_{i = 1}^{n}\; {A_{i}\varphi_{i}}}},{y + {\sum\limits_{i = 1}^{n}\; {A_{i}\eta_{i}}}},{t + 1}} )}},} & \; \\{I_{z}:={{I( {{x + {\sum\limits_{i = 1}^{n}\; {A_{i}\varphi_{i}}}},{y + {\sum\limits_{i = 1}^{n}\; {A_{i}\eta_{i}}}},{t + 1}} )} - {{I( {x,y,t} )}.}}} & \;\end{matrix}$

2.2 The Affine Over-Parameterization Model

The affine model is a good approximation of the flow in large regions inmany real world scenarios. We therefore first start with the affinemodel for our method. Note that we do not force the affine model overimage patches as in previously considered image registration techniques,and here each pixel has “its own” independent affine model parameters.The affine model has n=6 parameters,

$\begin{matrix}{{{\varphi_{1} = 1};\mspace{14mu} {\varphi_{2} = \hat{x}};\mspace{14mu} {\varphi_{3} = \hat{y}};}{{\varphi_{4} = 0};\mspace{14mu} {\varphi_{5} = 0};\mspace{14mu} {\varphi_{6} = 0};}{{\eta_{1} = 0};\mspace{14mu} {\eta_{2} = 0};\mspace{14mu} {\eta_{3} = 0};}{{{\eta_{4} = 1};\mspace{14mu} {\eta_{5} = \hat{x}};\mspace{14mu} {\eta_{6} = \hat{y}}},}} & (9) \\{{where},} & \; \\{{\hat{x} = \frac{\rho ( {x - x_{0}} )}{x_{0}}},} & (10) \\{{\hat{y} = \frac{\rho ( {y - y_{0}} )}{y_{0}}},} & \;\end{matrix}$

and x₀ and y₀ are half image width and height respectively. ρ is aconstant that has no meaning in an unconstrained optimization such asthe Lucas-Kanade method. In our variational formulation, ρ is aparameter which weighs the penalty of the x and y coefficients relativeto the coefficient of the constant term in the regularization. Anequivalent alternative is to add a different weight for each coefficientin (6).

2.3 The Rigid Motion Model

The optic flow of an object moving in rigid motion or of a static scenewith a moving camera is described by

u=−θ ₁+θ₃ {circumflex over (x)}+Ω ₁ {circumflex over (x)}ŷ−Ω₂(1+{circumflex over (x)} ²)+Ω₃ ŷ,

v=−θ ₂+θ₃ ŷ+Ω ₁(1+ŷ ²)−Ω₂ {circumflex over (x)}ŷ−Ω ₃ {circumflex over(x)},  (11)

where (θ₁, θ₂, θ₃)^(T) is the translation vector divided by the depthand (Ω₁,Ω₂,Ω₃)^(T) is the rotation vector. Here too, the number ofcoefficients is n=6. The coefficients A_(i) represent the translationand rotation variables, A₁=θ₁; A₂=θ₂; A₃=θ₃; A₄=Ω₁; A₅=Ω₂; A₆=Ω₃, andthe basis functions are φ₁−1; Φ₂=0; φ₃={circumflex over (x)};φ₄={circumflex over (x)}ŷ; φ₅=−(1+{circumflex over (x)}²); φ₆=ŷ andηn₁=0; η₂−1η₃=ŷ; η₄=1+ŷ²; η₅=−{circumflex over (x)}ŷ; andη₆=−{circumflex over (x)}. Similar constraints on optical flow of rigidmotion were first introduced by Adiv in (Adiv 1985). However, there theoptical flow is a preprocessing step to be followed by structure andmotion estimation, while our formulation uses the rigid motion model inthe optimality criterion of the optical flow estimation process. Usingthe rigid motion model directly in the optimality criterion waspreviously suggested by Sekkati and Mitiche (2003) where the functionalis explicitly written in terms of the model parameters. However, sincethe data term in (Sekkati and Mitiche 2003) is quadratic and uses thelinearized brightness constancy assumption, it is expected to be moresensitive to outliers and prone to errors for motion fields of largemagnitudes. Since the optical flow induced by camera rotation isindependent of the depth it has a more global nature and therefore onemay wish to penalize more severely for changes in rotation whenconsidering the smoothness term. This can be done by scaling the basisfunctions multiplying the rotation coefficients by a factor between 0and 1. Such a factor would require larger changes in the coefficients inorder to achieve the same overall influence of the rotation on theoptical flow. Such larger changes in the corresponding coefficientswould be suppressed by the regularization term and therefore achieve amore global effect of the rotation. Note, that assuming rigid motion onecould also extract the depth profile (up to scaling) from the abovecoefficients.

2.4 Pure Translation Motion Model

A special case of the rigid motion scenario can be thought of when welimit the motion to simple translation. In this case we have,

u=−θ ₁+θ₃ {circumflex over (x)},

v=−θ ₁+θ₃ ŷ,  (12)

The Euler-Lagrange equations of the rigid motion still applies in thiscase when considering only the first n=3 coefficients and correspondingbasis functions.

2.5 Constant Motion Model

The constant motion model includes only n=2 coefficients, with

φ₁=1; φ₂=0 η₁=0; η₂=1,  (13)

as basis functions. For this model there are two coefficients to solvefor A₁ and A₂ which are the optic flow components u and v, respectively.In this case we obtain the familiar variational formulation where wesolve for the u and v components. This fact can also be seen bysubstitution of (13) into (7), that yields the Euler-Lagrange equationsused, for example, in (Brox et al. 2004) (without the gradient constancyterm).

3. NUMERICAL SCHEME

In our numerical scheme we use a multi-resolution solver, by downsampling the image data with a standard factor of 0.5 along the x and yaxes between the different resolutions. The solution is interpolatedfrom coarse to fine resolution. Similar techniques to overcome theintrinsic non-convex nature of the resulting optimization problem wereused, for example, in (Brox et al. 2004), we compare our model andresults to (Brox et al. 2004) since, to the best of our knowledge, thatpaper reports the most accurate flow field results for the Yosemitewithout clouds sequence. At the lowest resolution, we start with theinitial guess of A_(i)=0, i=1, . . . , n. From this guess, the solutionis iteratively refined and the coefficients are interpolated to becomethe initial guess at the next higher resolution and the process isrepeated until the solution at the finest resolution is reached. At eachresolution, three loops of iterations are applied. At the outer loopwith iteration variable k, we freeze the brightness constancy linearapproximation in the Euler-Lagrange equations

$\begin{matrix}{{{{\Psi^{\prime}( ( I_{z}^{k + 1} )^{2} )} \cdot I_{z}^{k + 1} \cdot ( {{( I_{x}^{+} )^{k}\varphi_{q}} + {( I_{y}^{+} )^{k}\eta_{q}}} )} - {\alpha \cdot {{div}\begin{pmatrix}{\Psi^{\prime}( {\sum\limits_{i = 1}^{n}{{\overset{\sim}{\nabla}A_{i}^{k + 1}}}^{2}} )} \\{\hat{\nabla}A_{q}^{k + 1}}\end{pmatrix}}}} = 0.} & (14)\end{matrix}$

Inserting the first terms of the Taylor expansion

$\begin{matrix}{{I_{z}^{k + 1} \approx {I_{z}^{k} + {( I_{x}^{+} )^{k}{\sum\limits_{i = 1}^{n}{{dA}_{i}^{k}\varphi_{i}}}} + {( I_{y}^{+} )^{k}{\sum\limits_{i = 1}^{n}{{dA}_{i}^{k}\eta_{i}}}}}},} & (15)\end{matrix}$

where A_(i) ^(k+1)=A_(i) ^(k)+dA_(i) ^(k). The second inner loop—fixedpoint iteration—deals with the nonlinearity of Ψ with iteration variable1, it uses the expressions of Ψ′ from the previous iteration in both thedata and smoothness terms, while the rest of the equation is writtenwith respect to the l+1 iteration

$\begin{matrix}{{{{( \Psi^{\prime} )_{Data}^{k,l}{d_{q}\begin{pmatrix}{I_{z}^{k} +} \\{\sum\limits_{i = 1}^{n}{{dA}_{i}^{k,{l + 1}}d_{i}}}\end{pmatrix}}} - {\alpha \cdot {{div}( {( \Psi^{\prime} )_{Smooth}^{k,l}{\hat{\nabla}\begin{pmatrix}{A_{q}^{k} +} \\{dA}_{q}^{k,{l + 1}}\end{pmatrix}}} )}}} = 0},} & (16) \\{{where},} & \; \\{{d_{m}:={{( I_{x}^{+} )^{k}\varphi_{m}} + {( I_{y}^{+} )^{k}\eta_{m}}}},} & (17) \\{{( \Psi^{\prime} )_{Data}^{k,l}:={\Psi^{\prime}( \begin{pmatrix}{I_{z}^{k} +} \\{\sum\limits_{i = 1}^{n}{{dA}_{i}^{k,l}d_{i}}}\end{pmatrix}^{2} )}},} & (18) \\{and} & \; \\{( \Psi^{\prime} )_{Smooth}^{k,l}:={{\Psi^{\prime}( {\sum\limits_{i = 1}^{n}{{\overset{\sim}{\nabla}( {A_{i}^{k} + {dA}_{i}^{k,l}} )}}^{2}} )}.}} & (19)\end{matrix}$

At this point, we have for each pixel n linear equations with nunknowns, the increments of the coefficients of the model parameters.The linear system of equations is solved on the sequence volume usingGauss-Seidel iterations. Each Gauss-Seidel iteration involves thesolution of n linear equations for each pixel as described by (16). Thediscretization uses two point central difference for the flow componentsand four point central difference for the image derivatives as suggestedin (Barron et al. 1994).

4. EXPERIMENTAL RESULTS

In this section we compare our optical flow computation results to thebest published results. For test sequences with ground truth, we use thestandard measures of Average Angular Error (AAE) and Standard Deviation(STD). Our results are measured over all the pixels (100% dense). Theangle is defined as:

$\begin{matrix}{\theta = {\arccos ( \frac{{uu}_{g} + {vv}_{g} + 1}{\sqrt{( {u^{2} + v^{2} + 1} )( {u_{g}^{2} + v_{g}^{2} + 1} )}} )}} & (20)\end{matrix}$

where u and v are the estimated optical flow components, and u_(g) andv_(g) represent the ground truth optical flow. The AAE is the averageand STD is the standard deviation of θ over the image domain.

4.1 Parameter Settings

The parameters were set experimentally by an optimization process whichnumerically minimizes the weighted sum of AAE and STD measured on theYosemite sequence. Such a parameter optimization or training process isusual in many other papers, for example (Roth and Black 2005). Brox etal. (2004), Papenberg et al. (2006) also seem to have an optimalparameter setting since in their parameter sensitivity analysis, eachparameter change in any direction results in an increase of the AAEmeasure for the Yosemite sequence.

TABLE 1 Parameter settings Method α ρ ω_(t) ² ε σ Constant motion (3D)16.0 — 9.0 0.001 0.8 Affine (2D) 58.3 0.858 0 0.001 0.8 Pure translation(2D) 51.0 0.575 0 0.001 0.8 Affine (3D) 32.9 1.44 0.474 0.001 0.8 Rigidmotion (3D) 54.6 1.42 0.429 0.001 0.8 Pure translation (3D) 23.6 1.220.688 0.001 0.8

We have found slightly different parameter settings for the 2D and 3Dsmoothness cases as shown in Table 1, where 2D refers to smoothness termwith only spatial derivatives, while 3D refers to spatio-temporalsmoothness term that couples the solution of the optic flow field atdifferent frames (also known as the “two frames” versus “multi-frame”formulation). Here a denotes the standard deviation of the 2D Gaussianpre-filter used for pre-processing the image sequence. We used 60iterations of the outer loop in the 3D method and 80 iterations in the2D method, 5 inner loop iterations and 10 Gauss-Seidel iterations.

4.2 Yosemite Sequence

We applied our method to the Yosemite sequence without clouds (availableat http://www.cs.brown.edu/people/black/images.html), with fourresolution levels.

TABLE 2 Yosemite sequence without clouds Method AAE STD Papenberg et al.2D smoothness (Papenberg et al. 2006) 1.64° 1.43° Brox et al. 2Dsmoothness (Brox et al. 2004) 1.59° 1.39° Mémin and Pérez (2002) 1.58°1.21° Roth and Black (2005) 1.47° 1.54° Bruhn et al. (2005) 1.46° 1.50°Amiaz et al. 2D smoothness (Over-fine x4) 1.44° 1.55° (Amiaz et al.2007) Farnebäck (2000) 1.40° 2.57° Liu et al. (2003) 1.39° 2.83° Ourmethod affine 2D smoothness 1.18° 1.31° Govidu (2006) 1.16° 1.17°Farnebäck (2001) 1.14° 2.14° Our method constant motion 3D smoothness1.07° 1.21° Papenberg et al. 3D smoothness (Papenberg et al. 2006) 0.99°1.17° Brox et al. 3D smoothness (Brox et al. 2004) 0.98° 1.17° Ourmethod rigid motion 3D smoothness 0.96° 1.25° Our method affine 3Dsmoothness 0.91° 1.18° Our method pure translation 3D smoothness 0.85°1.18°

Table 2 shows our results relative to the best published ones. As seenin the table, our method achieves a better reconstructed solutioncompared to all other reported results for this sequence, both for the2D and 3D cases. In fact, our result for the 2D case is good evencompared to 3D results from the literature. FIGS. 4 and 5 show thesolution of the affine parameters from which one can observe the trendsof the flow changes with respect to the x and y axes. The solution ofthe pure translation parameters is shown in FIG. 6. The depthdiscontinuities in the scene are sharp and evident in all the parameters

FIG. 1 shows the ground truth optical flow and the results with the puretranslation model. FIG. 2 shows the image of the angular errors. FIG. 3shows both the histogram and the cumulative probability of the angularerrors. Both figures demonstrating the typically lower angular errors ofour method.

TABLE 3 Yosemite without clouds-noise sensitivity results presented asAAE ± STD Our method (affine) coupled Our method Results reported inσ_(n) with (Nir et al. 2005) (affine) (Brox et al. 2004) 0 0.91 ± 1.18°0.93 ± 1.20° 0.98 ± 1.17° 20 1.59 ± 1.67° 1.52 ± 1.48° 1.63 ± 1.39° 402.45 ± 2.29° 2.02 ± 1.76° 2.40 ± 1.71°

Table 3 summarizes the noise sensitivity results of our method. We alsocoupled the affine over-parameterized model with our previous work onjoint optic flow computation and denoising presented in (Nir et al.2005) by iterating between optical flow computation on the denoisedsequence and denoising with the current optical flow. A related conceptwas first introduced by Borzi et al. (2002). This coupling provides amodel with robust behavior under noise, that obtains better AAE measureunder all noise levels compared to the best published results.

TABLE 4 Yosemite sequence without clouds-results obtained by using equalspatio-temporal smoothness weights (ω_(t) = 1) Method AAE STD Our methodaffine 3D smoothness 0.93° 1.20° Our method pure translation 3Dsmoothness 0.86° 1.18°

Our improvement relative to (Brox et al. 2004) results mostly from theover-parameterization and not by the additional smoothness weightparameter co_(t). Table 4 shows the results obtained with equalspatio-temporal smoothness weights (co_(t)=1) as used in Brox et al.(2004). The AAE measure changes by approximately 1 percent for the puretranslation model and 2 percent for the affine model, where, the ω_(t)parameter was changed by approximately 21 and 45 percent for the puretranslation and affine models respectively.

4.3 Synthetic Piecewise Constant Affine Flow Example

For illustration purposes we also considered the piecewise affine flowover an image of size 100×100 having the ground truth shown in FIG. 8,and given by

For x<40,

u=−0.8−1.6(x−50)/50+0.8(y−50)/50,

v=1.0+0.65(x−50)/50−0.35(y−50)/50.

For x≧40,

u=0.48−0.36(x−50)/50−0.6(y−50)/50,

v=0.3−0.75(x−50)/50−0.75(y−50)/50.

The two images used for this test were obtained by sampling a grid ofsize 100×100 from frame 8 of the Yosemite sequence (denotedI_(yosemite)). The second image I₂(x, y)=I_(yosemite)(x+Δx, y+Δy) andthe first image is sampled at warped locations I₁(x,y)=I_(yosemite)(x+Δx+y, y+Δy+v) using bilinear interpolation. Theconstant shift values are: Δx=79 and Δy=69. The two images obtained aredisplayed in FIG. 7.

TABLE 5 Synthetic piecewise affine flow test Method AAE STD Ourimplementation of the method of (Brox et 1.48° 2.28 al. 2004) Our methodwith the affine flow model 0.88° 1.67

The results exhibited in Table 5 show that our method with the affineover-parametrization outperforms the method of (Brox et al. 2004). Thisis to be expected since the true flow is not piecewise constant and thesmoothness term in (Brox et al. 2004) penalizes for changes from theconstant flow model, whereas, the affine over-parametrizationmodelsolves the optimization problem in the (correct) affine space in whichit accurately finds the piecewise constant affine parameters solution ofthe problem, as shown in FIGS. 9 and 10. One can notice that thediscontinuity at pixel x=40 is very well preserved due to the effectiveedge preserving L₁ based optimization. The resulting optical flow shownin FIG. 11 accurately matches the ground truth.

4.4 Flower Garden Sequence

We also applied our method to the real image sequence “flower garden”.The results obtained by the 2D affine model are shown in FIG. 12. Wehave checked our results by manually marking the coordinates of severalcorresponding points on the two images and comparing the motion with theoptical flow results. We have found good match between manual trackingand the computed flow. The tree moves with a velocity of about −5.7pixel/frame along the x direction. The whole scene has low velocity inthe y direction (the computed v is between −0.9 and 1.4 pixel/frame).

The y component of the flow is about 1 pixel/frame in the upper sectionof the tree and almost zero in the lower section. In the area of thegarden, the velocity is decreasing as the distance from the cameraincreases from about −3 pixel/frame in the right lower section (−2 onthe left) to about −1 pixel/frame in the upper part and in the area ofthe houses. The computed flow by the pure translation over-parameterizedmodel shown in FIG. 13 produces similar flow field as the results withthe affine model. The solution of the three pure translation modelparameters is shown in FIG. 14.

4.5 Road Sequence

The two frames of this real sequence courtesy of the authors of (Vázquezet al. 2006) are shown in FIG. 15. The computed flow by our 2Dover-parameterized affine model is shown in FIG. 16. The motioncompensated image difference is shown in FIG. 17, our motion compensatedreconstruction ratio is PSNR=38.17 dB (more than 1.13 dB better than theresults reported in (Vázquez et al. 2006) for the affine model and 0.58dB better than their quadratic model). The measure of the motioncompensated difference might be misleading for comparing the quality ofdifferent optical flow algorithms since one can generate a globallyoptimal algorithm for this measure which produces meaningless opticalflow results. For example: suppose we find for each pixel in one image acorresponding pixel with the same gray value in the other image. In thiscase, the resulting optical flow is broadcasts (resolutions such as852×576, 1,280×720, 1,920×1,080 or 1,920×1,080). Another applicationwould be to enhance the quality of a low-resolution video segment inorder to better recognize its contents. When increasing the resolutionof an image or a video segment, new information (pixels) needs to beadded that was not in the image before processing. By analyzing themovement of objects in the segment, the optical flow results accordingto the invention help complete the information that is to be representedin the added pixels.

-   -   Video compression—digital video segments can be voluminous and        typically need to be compressed for broadcasting and storage        purposes. Image sequence compression algorithms commonly use        optical flow parameters to represent images compactly in terms        of changes relative to preceding or succeeding images in the        sequence. As with any compression, the results can be an        approximation of the original video segment (lossy compression)        or an exact decompression. The majority of commercial video        compressions are lossy assuming that the human eye does need        absolutely all the information on an image in order for the        results to achieve good perceptual quality.    -   Object tracking—object tracking has many useful applications        such as: security and surveillance—to recognize people; medical        therapy—recognize movement patterns for physical therapy        patients and disabled people; retail—analyze shopping behavior        of customers; traffic management—analyze flow etc. Optical flow        analysis is a very useful tool for recognizing and then tracking        an object along multiple images (frames).    -   3D object reconstruction—it can be helpful to reconstruct        three-dimensional objects from two-dimensional camera images,        for example, by filming the same object from two (or more)        different angles (stereo vision, similar to the human eye stereo        vision). One major challenge of the reconstruction problem is to        find feature correspondences, that is, to locate the projections        of the same three-dimensional geometrical or textural feature on        two or more images. Comparing the optical flow information from        different video inputs (angles) can help recognize the same        object in the different video inputs, and thus enhance the        information collected about the object and its behavior.    -   Medical imaging—medical imaging applications benefit from        optical flow computations in several areas, for example,        reconstructing a 3D image of an area based on several images        taken from different angles.

Although the invention has been described in detail, neverthelesschanges and modifications, which do not depart from the teachings of thepresent invention, will be evident to those skilled in the art. Suchchanges and modifications are deemed to come within the purview of thepresent invention and the appended claims.

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1. A method of determining the optical flow vector of a plurality ofimage pixels between two consecutive images in a group of images, themethod comprising the steps of: (i) creating multiple sets of said groupof images wherein each set contains a decreasing number of pixels perimage; (ii) defining an optical flow motion model applicable to thegroup of images in terms of a set of basis functions; (iii) representingthe optical flow vector of each pixel image by different coefficients ofthe same motion model; (iv) starting with the set of group of imageswith the lowest number of pixels per image, determining respectiveover-parameterized optical flow vectors at the pixels responsively tomotion between the images in the group by computing respectiveregularized coefficients of the basis functions at each of the pixels;(v) interpolating the results to the next higher resolution set of groupof images and refining the estimation using the over-parameterizedoptical flow representation; and (vi) repeating step (v) until thehighest resolution set of group of images is reached.
 2. A methodaccording to claim 1, wherein said motion model is an affine motionmodel.
 3. A method according to claim 1, wherein said motion model is arigid motion model.
 4. A method according to claim 1, wherein said basisfunctions are spatial basis functions.
 5. A method according to claim 1,wherein said basis functions are spatio-temporal basis functions.
 6. Amethod according to claim 1, wherein a new group of images is createdsuch that each image in the new group of images contains more pixelsthan the corresponding image in the original group of images.
 7. Amethod according to claim 1, wherein the over-parameterized optical flowrepresentation is used within a super-resolution scheme.
 8. A methodaccording to claim 1, wherein the over-parameterized optical flowrepresentation is used to track a moving object in said group of images.9. A method according to claim 1, wherein the over-parameterized opticalflow representation is used in a video compression and/or decompressionapplication.
 10. A method according to claim 1, wherein theover-parameterized optical flow representation is used in a segmentationapplication.
 11. A computer-readable medium encoded with a programmodule that determines the optical flow vector of a plurality of imagepixels between two consecutive images in a group of images, by: (i)creating multiple sets of said group of images wherein each set containsa decreasing number of pixels per image; (ii) defining an optical flowmotion model applicable to the group of images in terms of a set ofbasis functions; (iii) representing the optical flow vector of eachpixel image by different coefficients of the same motion model; (iv)starting with the set of group of images with the lowest number ofpixels per image, determining respective over-parameterized optical flowvectors at the pixels responsively to motion between the images in thegroup by computing respective regularized coefficients of the basisfunctions at each of the pixels; (v) interpolating the results to thenext higher resolution set of group of images and refining theestimation using the over-parameterized optical flow representation; and(vi) repeating step (v) until the highest resolution set of group ofimages is reached.
 12. A medium according to claim 11, wherein saidmotion model is an affine motion model.
 13. A medium according to claim11, wherein said motion model is a rigid motion model.
 14. A mediumaccording to claim 11, wherein said basis functions are spatial basisfunctions.
 15. A medium according to claim 11, wherein said basisfunctions are spatio-temporal basis functions.
 16. A medium according toclaim 11, wherein a new group of images is created such that each imagein the new group of images contains more pixels than the correspondingimage in the original group of images.
 17. A medium according to claim11, wherein the over-parameterized optical flow representation is usedwithin a super-resolution scheme.
 18. A medium according to claim 11,wherein the over-parameterized optical flow representation is used totrack a moving object in said group of images.
 19. A medium according toclaim 11, wherein the over-parameterized optical flow representation isused in a video compression and/or decompression application.
 20. Amedium according to claim 11, wherein the over-parameterized opticalflow representation is used in a segmentation application.